On covering paths with 3 dimensional random walk

Eviatar B. Procaccia; Yuan Zhang

doi:10.1214/18-ECP160

On covering paths with 3 dimensional random walk

Eviatar B. Procaccia
, Yuan Zhang

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we find an upper bound for the probability that a 3 dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an L₁ ball of radius N in ℤ^d. For d ≥ 4, it has been shown in [5] that such probability decays exponentially with respect to N. For d = 3, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: ∀ε > 0, ∃c_ε > 0, (Formula Presented).

Original language	English
Article number	57
Journal	Electronic Communications in Probability
Volume	23
DOIs	https://doi.org/10.1214/18-ECP160
State	Published - 2018
Externally published	Yes

Keywords

3 dimensional random walk
Covering probability

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/18-ECP160

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title = "On covering paths with 3 dimensional random walk",

abstract = "In this paper we find an upper bound for the probability that a 3 dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an L1 ball of radius N in ℤd. For d ≥ 4, it has been shown in [5] that such probability decays exponentially with respect to N. For d = 3, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: ∀ε > 0, ∃cε > 0, (Formula Presented).",

keywords = "3 dimensional random walk, Covering probability",

author = "Procaccia, \{Eviatar B.\} and Yuan Zhang",

note = "Funding Information: *Texas A\&M University www.math.tamu.edu/\textasciitilde{}procaccia E-mail: [email protected] †Research supported by NSF grant DMS-1407558 ‡Peking University E-mail: [email protected] §The manuscript of this paper was first done when YZ was a visiting assistant professor at Texas A\&M University Publisher Copyright: {\textcopyright} 2018, University of Washington. All rights reserved.",

year = "2018",

doi = "10.1214/18-ECP160",

language = "English",

volume = "23",

journal = "Electronic Communications in Probability",

issn = "1083-589X",

publisher = "Institute of Mathematical Statistics",

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T1 - On covering paths with 3 dimensional random walk

AU - Procaccia, Eviatar B.

AU - Zhang, Yuan

N1 - Funding Information: *Texas A&M University www.math.tamu.edu/~procaccia E-mail: [email protected] †Research supported by NSF grant DMS-1407558 ‡Peking University E-mail: [email protected] §The manuscript of this paper was first done when YZ was a visiting assistant professor at Texas A&M University Publisher Copyright: © 2018, University of Washington. All rights reserved.

PY - 2018

Y1 - 2018

N2 - In this paper we find an upper bound for the probability that a 3 dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an L1 ball of radius N in ℤd. For d ≥ 4, it has been shown in [5] that such probability decays exponentially with respect to N. For d = 3, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: ∀ε > 0, ∃cε > 0, (Formula Presented).

AB - In this paper we find an upper bound for the probability that a 3 dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an L1 ball of radius N in ℤd. For d ≥ 4, it has been shown in [5] that such probability decays exponentially with respect to N. For d = 3, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: ∀ε > 0, ∃cε > 0, (Formula Presented).

KW - 3 dimensional random walk

KW - Covering probability

UR - https://www.scopus.com/pages/publications/85053664951

U2 - 10.1214/18-ECP160

DO - 10.1214/18-ECP160

M3 - Article

SN - 1083-589X

VL - 23

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

M1 - 57

ER -

On covering paths with 3 dimensional random walk

Abstract

Keywords

ASJC Scopus subject areas

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